User talk:Paul Martin/Hexadecimal metric system/Archive 02

Also see:  SMH talk: Archive I and Hexadecimal metric system.

The new digital foot

The hexadecimal foot compared to the Roman foot, Austrian foot (i.e. the Greek "pous metrios") and to the English and French foot measures :

The English foot, after 1066, is defined 16/15 French foot. The Roman foot and the English foot entertain the ratio 36 to 35 exactly one. An Austrian foot is 16/15 Roman foot. The new digital foot is always  48/49 English foot, but contests the current English foot definition. Definitively : 1575 French feet equal 1620 Austrian feet equal 1680 English feet equal 1715 digital feet equal 1728 Roman feet. The comparative distance is  512, 068 371, 171 875 nm with the scientific value; slightly shorter with the former partial, national definitions.
		Comparison distances (all below values in millimetres)
		1575 x 9 / 0.27706 ≥  511 622.031	1620 x 1896.5 / 6 =  512 055	1680 x 304.8 =  512 064	1715 x 19 109.257 / 64 =  512 068.371 171 875	1728 x 296.352 =  512 096.256
Ratio	Foot	French definition (1799)	Austrian definition (1872)	Anglo-saxon definition (1959)	Scientific definition (1990)	Idealistic definition (2003)
1575	French	≤  324.839 385	≤  325.114 286	=  325.120 000	=  325.122 775 347 222 222 222 222	=  325.140 480
1620	Austrian	≤  315.816 069	=  316.083 333	=  316.088 888	=  316.091 587 143 132 716 049 382	=  316.108 800
1680	English	≥  304.536 923	≤  304.794 643	=  304.800 000	=  304.802 601 888 020 833 333 333	=  304.819 200
1715	Digital	≥  298.321 884	≥  298.574 344	≤  298.579 592	=  298.582 140 625 000 000 000 000	=  298.598 400
1728	Roman	≥  296.077 564	=  296.328 125	=  296.333 333	=  296.335 862 946 686 921 296 296	=  296.352 000
Percentages :		≤ 99.912 836 %	≤ 99.997 389 %	≥ 99.999 146 %	=  100.000 000 000 %	≥ 100.005 445 %
The new hexadecimal foot has a modern, scientific definition. From its definition, scientific values for the four other feet are deduced. The new worldwide hexadecimal foot has been conceived in A.D. 1989. Since 1990, the value of this foot is 19.109 257 m / 64 exactly one. This value of the digital foot is obtained by the adequate definition: Math.round ( r0 * Math.PI * Math.pow(10,6) / Math.pow(16,5) ) / 64. The radius r0 is the World Geodetic System 1984 value of the major radius of our planet Earth, scilicet:  6, 378 137 metres  ± 1 m. Thus, the 64-digital-feet chain is very flimsy over-defined, since ± 1 m values almost  ± 3 µm in this chain definition.  However:  A definition is a definition ! The idealistic definition of 2003 is merely a licence, outside of all scientific and extremely high-level accuracy fields. An idealistic digital foot of 1728 x 0.1728 = 298.5984 mm gives easy values for all the other historical foot measures. The anglo-saxon "compromise foot" of 1959  – however very close to the scientific value –  is not well-defined! The percentages give the slight errors of the former partial, national definitions. The difference in the idealistic definition is assumed for always obtain simple prime factors.

Ratios between these five foot measures, notably their linear relationship to the new digital foot.

Foot	French	Austrian	English	Digital	Roman	Idealistic value in mm
French	1 : 1	35 : 36	15 : 16	45 : 49	175 : 192	28  x  34  x  5-5  x  72
Austrian	36 : 35	1 : 1	27 : 28	324 : 343	15 : 16	26  x  32  x  5-4  x  73
English	16 : 15	28 : 27	1 : 1	48 : 49	35 : 36	24  x  35  x  5-4  x  72
Digital	49 : 45	343 : 324	49 : 48	1 : 1	1715 : 1728	28  x  36  x  5-4  x  70
Roman	192 : 175	16 : 15	36 : 35	1728 : 1715	1 : 1	22  x  33  x  5-3  x  73

 The exact value of the scientific 64-foot-chain is by its definition of 1990: 19 109 257 micrometres. This value is a huge prime number.
 Therefore, the scientific and idealistic values entertain the ratio:  (19 109 257 x 10)  :  (1728 x 1728 x 64),  a factor of  ≈ 0.999 945 5.